On Deciding Topological Classes of Deterministic Tree Languages

  • Filip Murlak
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3634)


It has been proved by Niwiński and Walukiewicz that a deterministic tree language is either Π\(_{\rm 1}^{\rm 1}\)-complete or it is on the level Π\(_{\rm 3}^{\rm 0}\) of the Borel hierarchy, and that it can be decided effectively which of the two takes place. In this paper we show how to decide if the language recognized by a given deterministic tree automaton is on the Π\(_{\rm 2}^{\rm 0}\), the Σ\(^{\rm 0}_{\rm 2}\), or the Σ\(^{\rm 0}_{\rm 3}\) level. Together with the previous results it gives a procedure calculating the exact position of a deterministic tree language in the topological hierarchy.


deterministic tree automata index hierarchy Borel hierarchy 


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  1. 1.
    Bradfield, J.C.: The modal mu-calculus alternation hierarchy is strict. Theoret. Comput. Sci. 195, 133–153 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Browne, A., Clarke, E.M., Jha, S., Long, D.E., Marrero, W.: An improved algorithm for the evaluation of fixpoint expressions. Theoret. Comput. Sci. 178, 237–255 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Emerson, E.A., Jutla, C.S.: The complexity of tree automata and logics of programs. In: Proc. FoCS 1988, pp. 328–337. IEEE Computer Society Press, Los Alamitos (1988)Google Scholar
  4. 4.
    Jurdziński, M., Vöge, J.: A discrete strategy improvement algorithm for solving parity games. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 202–215. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  5. 5.
    Kechris, A.S.: Classical Descriptive Set Theory. Graduate Texts in Mathematics, vol. 156. Springer, Heidelberg (1995)zbMATHGoogle Scholar
  6. 6.
    Kupferman, O., Safra, S., Vardi, M.: Relating Word and Tree Automata. In: 11th IEEE Symp. on Logic in Comput. Sci., pp. 322–332 (1996)Google Scholar
  7. 7.
    Lenzi, G.: A hierarchy theorem for the mu-calculus. In: Meyer auf der Heide, F., Monien, B. (eds.) ICALP 1996. LNCS, vol. 1099, pp. 87–109. Springer, Heidelberg (1996)Google Scholar
  8. 8.
    Mostowski, A.W.: Hierarchies of weak automata and weak monadic formulas. Theoret. Comput. Sci. 83, 323–335 (1991)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Niwiński, D.: On fixed point clones. In: Kott, L. (ed.) ICALP 1986. LNCS, vol. 226, pp. 464–473. Springer, Heidelberg (1986)Google Scholar
  10. 10.
    Niwiński, D., Walukiewicz, I.: Relating hierarchies of word and tree automata. In: Meinel, C., Morvan, M. (eds.) STACS 1998. LNCS, vol. 1373, pp. 320–331. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  11. 11.
    Niwiński, D., Walukiewicz, I.: A gap property of deterministic tree languages. Theoret. Comput. Sci. 303, 215–231 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Niwiński, D., Walukiewicz, I.: Deciding nondeterministic hierarchy of deterministic tree automata. In: Proc. WoLLiC 2004(2004) (to appear in Electronic Notes in Theoretical Computer Science)Google Scholar
  13. 13.
    Otto, M.: Eliminating recursion in μ-calculus. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 531–540. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  14. 14.
    Rabin, M.O.: Decidability of second-order theories and automata on infinite trees. Trans. Amer. Soc. 141, 1–35 (1969)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Seidl, H.: Fast and simple nested fixpoints. Information Processing Letters 59, 303–308 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Skurczyński, J.: The Borel hierarchy is infinite in the class of regular sets of trees. Theoret. Comput. Sci. 112, 413–418 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Thomas, W.: Languages, automata, and logic. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 3, pp. 389–455. Springer, Heidelberg (1997)Google Scholar
  18. 18.
    Urbański, T.F.: On deciding if deterministic Rabin language is in Büchi class. In: Welzl, E., Montanari, U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853, pp. 663–674. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  19. 19.
    Wagner, K.: Eine topologische Charakterisierung einiger Klassen regulärer Folgenmengen. J. Inf. Process. Cybern. EIK 13, 473–487 (1977)zbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Filip Murlak
    • 1
  1. 1.Institute of InformaticsWarsaw UniversityWarszawaPoland

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